Although I can't completely rule out the possiblity that someone might
be able to train themselves to sneeze with their eyes open, I am inclined
to think you wouldn't be able to do it.

It's not too difficult to use conditioning techniques to modify
reflexes, in the sense of making them stronger or weaker, or easier or
harder to evoke. It's much more difficult, and in some cases impossible, to
change the reflex itself (add or remove a component). This is particularly
true of simple reflexes.

However, a sneeze is a pretty complex reflex. (You can find several
posts that describe it in more detail in the Mad Scientist's archive; just
search under "sneeze". ) Since it is neurally complex, theoretically it
may be possible to use something called instrumental conditioning to modify
the reflex.

Instrumental conditioning is essentially learning that a particular
behavior predicts a particular outcome. Here's an example. Normally, rats
don't go around pressing levers. However, if a lever is around, a rat may,
for some reason, press it once in awhile. If the rat is hungry, and you
give him food every time he presses the lever, pretty soon he'll be
pressing the lever alot (at least until he's not hungry anymore). The rat
has been instrumentally conditioned to press the lever.

One thing you can do with instrumental conditioning is shape behavior.
For example, when the rat was first learning that moving the lever gets him
food, you might reward him if he only touches the lever. After he begins
touching the lever regularly, you might require him to move it a tiny bit
to get food. By changing what the rat has to do to get his reward a bit at
a time, eventually you can get the rat to push the lever very far, or maybe
in a particular direction. You have shaped his behavior.

You probably see where this is leading. You might be able to use a
similar procedure to shape a sneeze. It would be tough though. First, you
need to have a way to reliably get yourself to sneeze. Then you need a way
to monitor how much your eyes close, and hope that sometimes they don't
close quite as much as other times, so you have something to work with at
the beginning. Then you need to find some sort of reinforcement that will
tend to pressure the behavior to change (a little electric shock every time
your eyes close completely comes to mind!). Finally, you have to hope that
the form of a sneeze is something that can be modified by learning. In any
event, it would take an awful lot of time and energy.